Real solutions of the first Painlevé equation with large initial data

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1 Real solutions of the first Painlevé equation with large initial data arxiv:6.00v math.ca] Jun 07 Wen-Gao Long, Yu-Tian Li, Sai-Yu Liu and Yu-Qiu Zhao Department of Mathematics, Sun Yat-sen University, GuangZhou 07, China Department of Mathematics, and Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon, Hong Kong School of Mathematics and Computation Science, Hunan University of Science and Technology, Xiangtan 40, China Abstract We consider three special cases of the initial value problem of the first Painlevé equation PI. Our approach is based on the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod. A rigorous proof of a property of the PI solutions on the negative real axis, recently revealed by Bender and Komijani, is given by approximating the Stokes multipliers. Moreover, we build more precise relation between the large initial data of the PI solutions and their three different types of behavior as the independent variable tends to negative infinity. In addition, some limiting form connection formulas are obtained. Keywords: Connection formula; first Painlevé transcendent; uniform asymptotics; Airy function; modified Bessel function; Stokes multiplier MSC00: 4M40, E7, 4A, 4E0, C0 Introduction and main results The first Painlevé equation has the canonical form d y dt = 6y + t.. It is known that there are two kinds of solutions of PI, behaving respectively as yt t t and yt 6 6 Corresponding author Yu-Qiu Zhao. address: stszyq@mail.sysu.edu.cn

2 as t ; see for example Bender and Orszag ]. Refinements have been obtained for the t second case, such that the solutions oscillate stably about the parabola y =, with 6 where y = t + d t 8 cos φt + O t 8 6 φt = 4 4 as t,. ] 4 t 4 8 d log t + θ,. and d > 0 and θ are constants; cf. Kapaev ], Joshi and Kruskal ], and Qin and Lu 9], see also. of the handbook 8]. Moreover, numerical analysis conducted in Holmes and Spence 0], Fornberg and Weideman 9] and Qin and Lu 9] indicates that there exist constants κ < 0 and κ > 0, such that all solutions of. with y0 = 0 and κ < y 0 < κ belong to the second kind, while otherwise, if y 0 > κ or y 0 < κ, the solutions will blow up on the negative real axis. Therefore, it is natural to consider the initial value problem of. with initial data y0 = a, y 0 = b..4 The problem is how to determine the asymptotic behavior of yt with a given real pair a, b. In particular, if yt t/6 as t, how to establish the relation between the initial data.4 and the parameters d and θ in.-.. This is an open problem mentioned by Clarkson in several occasions 4, ]. In fact, before Clarkson s open problem is being proposed, investigations 0, 6] have been made on the PI functions on the negative real axis. In 0], a boundary value problem for PI is studied by Holmes and Spence, and it is shown that there are exactly three types of real solutions of PI equation. Later on, Kapaev ] obtained the asymptotic behavior of these solutions as follows: A a two-parameter family of solutions, oscillating about the parabola y = t/6 and satisfying. as t, where 4 4 d = π log s 0, 4 4 θ = arg s 4 4 d 9 8 log + 8 log π4 arg Γ i 4 4 d ;. B a one-parameter family of solutions termed separatrix solutions, satisfying t yt = 6 h { 4 π 4 8 t 8 exp t 4 + O t.6 as t, where h = s s 4 ;.7

3 C a two-parameter family of solutions, having infinitely many double poles on the negative real axis and satisfying { 6 yt + t/6 sin 4/4 t ρ log t + σ as t,.8 where ρ = π log s = π log + s s = σ = 9 8 ρ log + 8 ρ log + arg Γ iρ π log s 0, π 4 + arg s..9 In the above formulas, s k for k = 0,,,, 4 are the Stokes multipliers associated with the given solution; see the definition of Stokes multipliers in.4 below. Moreover, using the isomonodromy condition, Kapaev ] also got the following necessary conditions for the Stokes multipliers of each solution type: + s s > 0 for type A, + s s = 0 for type B, + s s < 0 for type C..0 It should be noted that the solutions of types A and B may also have finite poles on the negative real axis; see, Figures and ]. In the present paper, we focus on three special cases of the initial value problem.4 of the PI equation.. More precisely, we shall establish the relation between the asymptotic behavior of the PI solution yt as t and the real initial data y0, y 0 = a, b, in the following cases: I fixed a and large positive or, negative b; II fixed b and large negative a; III fixed b and large positive a. The motivation of the first two cases comes from a recent investigation by Bender and Komijani ], in which the unstable separatrix solutions of PI on the negative real axis are studied numerically and analytically. For case I, Bender and Komijani conclude that for any fixed initial value y0, there exists a sequence of initial slopes y 0 = b n that give rise to separatrix solutions. Figures and in ] indicate an interesting phenomenon of the PI solutions as the initial slope varies. It seems that, when b n < y 0 < b n+, the solution passes through n double poles and then oscillates stably about t/6, while for b n < y 0 < b n, the solution has infinitely many double poles. Moreover, they establish the asymptotic behavior πγ b n Bn n = 6 Γ ] as n +. by studying the eigenvalue problem associated with the PT -symmetric Hamiltonian Ĥ = ˆp + iˆx ; see ]. They have also obtained similar results for case II. However, part of their

4 argument is based on numerical evidences see, p.8]. To give a rigorous proof of their results and to build a precise relation between the initial data and the large negative t asymptotics, further analysis is needed. Other than the above two cases, case III is essentially different, and has not been addressed before, as far as we are aware of. In the above cases I-III, we consider the initial value problem by using the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod ], and further developed by Wong and Zhang, ], Zeng and Zhao ], and Long, Zeng and Zhou 7]. Our main results are stated as follows. Parts of Theorems and are given in ] and the rest are new. First, for case I, we shall prove the following results. Theorem. Suppose that yt; a, b is the real solution of. with initial data y0, y 0 = a, b, then for any fixed a, there exists an ascending sequence, 0 < b < b < < b n <, such that yt; a, b n are separatrix solutions, namely, of type B, with b n possessing the large-n asymptotic behavior., and the quantity h in.7 having the approximation { bn / h := h n = n exp P 0 + o as n +.. Moreover, i if b m < b < b m+, m =,,, then yt; a, b belongs to type A and { 4 4 d = b ] b P0 log cos Q0 Q + o, π 4 b 4 θ = Q0 Q 4 4 d 9 8 log + 8 log + π 4 arg Γ i 4 4 d + o as b + ; ii if b m < b < b m, m =,,, then yt; a, b belongs to type C and { ρ = b ] b P0 log cos Q0 Q + o, π σ = 9 8 ρ log + 8 ρ log + arg Γ as b +. iρ π b + Q0 Q + o..4 In.,. and.4, the constants P 0, Q 0 and Q are expressible in terms of the beta function as P 0 = 6 B,, Q 0 = B,, Q = π.. There also exists a descending sequence, < b n < < b < b < 0, having the similar properties as those of {b n. 4

5 For case II, the following holds. Theorem. Suppose that yt; a, b is the real solution of. with initial data y0, y 0 = a, b, then for any real fixed b, there exists a negative descending sequence 0 > a > a > > a n >, such that yt; a n, b are separatrix solutions, namely, of type B, with a n possessing the asymptotic behavior a n πγ/6 Γ/ ] n as n +, and the quantity h in.7 having the approximation h := h n = n exp { a n / P 0 + o as n +..6 Moreover, i if a m+ < a < a m, m =,,, then yt; a, b belongs to type A and 4 4 d = { ] a P0 log cos a Q0 + o, π 4 4 θ = a Q0 4 4 d 9 8 log + 8 log + π 4 arg Γ i 4 4 d + o.7 as a ; ii if a m < a < a m, m =,,, then yt; a, b belongs to type C and ρ = { ] a P0 log cos a Q0 + o, π as a. σ = 9 8 ρ log + 8 ρ log + arg Γ iρ π + a Q0 + o In.6,.7 and.8, the constants P 0 and Q 0 are the same as the ones in.. Case III is in a sense different from the first two cases. We have the following result..8 Theorem. Suppose that yt; a, b is the real solution of. with initial data.4. For any fixed b, there exists a positive number M depending on b such that, if a > M, then yt; a, b belongs to type C, and where H 0 = B, 6. ρ = ] a H0 + o, π σ = 9 8 ρ log + 8 ρ log + arg Γ iρ a H0 + o,.9

6 For the sake of convenience, we recall some important concepts in the isomonodromy theory for the first Painlevé transcendents. First, one of the Lax pairs for the PI equation is given as follows see 4]: { Ψ λ = 4λ 4 + t + y σ i4yλ + t + y σ y t λ + λ σ Ψ = AλΨ, { Ψ t = λ + y λ σ iy.0 λ σ Ψ = BλΨ, where σ = 0, σ 0 = 0 i 0, σ i 0 = 0 are the Pauli matrices and y t = dy. Compatibility of the above system implies that y = yt dt satisfies the first Painlevé equation.. Under the transformation Φλ = λ 4 σ σ + σ Ψ λ,. the first equation of.0 becomes Φ λ = yt λ + yλ t + y Φ.. λ y y t Following 4] see also ], the only singularity of equation. is the irregular singular point at λ =, and there exist canonical solutions Φ k λ, k Z, of. with the following asymptotic expansion Φ k λ, t = λ 4 σ σ + σ I + H λ + O e 4 λ λ +tλ σ, λ, λ Ω k,. where H = y t y tyσ, and the canonical sectors are { Ω k = λ C : arg λ π + kπ, π + kπ, k Z. These canonical solutions are related by Φ k+ = Φ k S k, S k = sk, S 0 k = 0,.4 s k where s k are called Stokes multipliers, and independent of λ and t according to the isomonodromy condition. The Stokes multipliers are subject to the constraints s k+ = s k and s k = i + s k+ s k+, k Z.. Moreover, regarding s k as functions of t, yt, y t, they also satisfy, p.687, ] s k t, yt, y t = s k t, yt, y t, k Z,.6 6

7 where ζ stands for the complex conjugate of a complex number ζ. From., it is readily seen that, in general, two of the Stokes multipliers determine all others. The derivation of.,.4 and., and more details about the Lax pairs, are referred to in 8]. The rest of the paper is arranged as follows. In Sec., we prove the main theorems, assuming the validity of Lemmas, and. Next, in Sec., we apply the method of uniform asymptotics to calculate the Stokes multipliers when t = 0, and hence to prove the three lemmas. In the last section, Sec. 4, a discussion is provided on prospective studies and possible difficulties in the general case of Clarkson s open problem. Some technical details are put in Appendices A, B, C and D to clarify the derivation. Proof of the main results Because the monodromy data {s 0, s, s, s, s 4 and the solutions of PI have a one-to-one correspondence, a general idea to solve connection problems is to calculate the Stokes multipliers of a specific solution in the two specific situations to be connected. In the initial data case, this means to calculate all s k as t and at t = 0. When t, as stated above in.,.7 and.9, the Stokes multipliers have been derived by Kapaev ]. When t = 0, it is much difficult to find the exact values of s k. However, inspired by the ideas in Sibuya 0], we are able to obtain their asymptotic approximations in the special cases considered here, as a step forward. These approximations, together with.,.7,.9 and.0, suffice to prove our theorems. First, the following lemma is crucial to prove Theorem. Lemma. For any fixed a and large positive or negative b, the asymptotic behaviors of the Stokes multipliers, corresponding to yt; a, b, are given by { b s 0 = i exp { b s = i exp { b s = i exp { E0 + F 0 cos b E0 sgnb E + o Q0 sgnb Q ] = s 4, + o E0 F 0 sgnb E F + o = s as b +, where sgnb is the sign of b and E 0 = F 0 = B, i B,, E = F = πi 6, Q 0 = ie 0 F 0 = B,., Q = ie F = π.,. An immediate consequence of. is stated as follows. 7

8 Corollary. For any fixed a, there exists a positive sequence {b n and a negative sequence { b n, n =,,, with nπ π/ + Q b n Q 0 and nπ π/ Q bn,. Q 0 as n, such that the Stokes multipliers, corresponding to yt; a, b n or yt; a, b n, satisfy s 0 = 0, s = s = s + s 4 = i. n 4 0 the true values of b n Bn nπ π/+q Q 0 / Table : When a = 0, the comparison of the approximate values.,. and the true values of b n, using five significant digits. The true values of b n are taken from ]. Proof of Theorem. We only give the proof when b > 0. When b < 0, the argument is the same except for some minor justification of signs; see.. According to.0, see also, Theorems. and.], regarding s 0 as a function of a and b, then s 0 a, b n = 0 implies that the solutions yt; a, b n all belong to type B. Moreover,. implies., and is an improved approximation of b n. Although both formulas are only applicable for large n,. has a better accuracy for small n when a = 0 as compared with.; see Table. For fixed a, from., it is obvious that the sequence {b n can be chosen such that + s s = Im s 0 a, b { > 0, if bm < b < b m+, < 0, if b m < b < b m, m =,,. It means that if b varies in the above two sequences of open intervals, the PI solutions yt; a, b correspond alternatively to type A and type C solutions. To get., we calculate the leading asymptotic behavior of s s 4 as b +. In fact, as a consequence of., we get { b s s 4 = i exp { b = exp { b = exp E0 E + o E0 + F 0 P0 { sin { sin i i exp { b F0 F + o b E0 F 0 ie F b Q0 Q + o + o.4 as b +. Here we have set P 0 = E 0 +F 0. Putting b = b n, and noting that b n / Q0 Q = nπ π + o as n, we obtain.. 8

9 Next, to obtain the limiting form connection formulas in., we only need to calculate s 0 and arg s. According to. and., one has { { / / b b s 0 = exp E 0 + F 0 cos Q 0 Q + o,. arg s = π b / Q 0 + Q + o as b +. Substituting. into. and noting that P 0 = E 0 + F 0, one may immediately get.. Finally, to obtain.4, we calculate s 0 and arg s, which can also be derived from.. This completes the proof of Theorem. Similarly, based on the following result, we can prove Theorem. Lemma. For fixed b and large negative a, the asymptotic behaviors of the Stokes multipliers, corresponding to yt; a, b, are given by s 0 = i exp { a / E 0 + F 0 { cos a / Q 0 ] + o, s = i exp { a / E 0 + o = s 4, s = i exp { a / E 0 F 0 + o.6 = s as a, where Q 0, E 0 and F 0 are the same constants as the ones in.. Corollary. There exists a negative discrete set {a n, n =,,, with nπ π a n Q 0 as n,.7 such that the Stokes multipliers corresponding to yt; a n, b are s 0 = 0, s = s = s + s 4 = i. Furthermore, Theorem is an immediate consequence of a combination of Lemma and.0. Lemma. For fixed b and large positive a, the asymptotic behaviors of the Stokes multipliers, corresponding to yt; a, b, are given by { s 0 = i exp a H0 + o, { s = i exp a G0 + o = s 4,.8 { s = i exp a G0 + H 0 + o = s as a +, where H 0 = B, 6 and G0 = B, 6 + i B, πi 6 = e H 0. We leave the proof of Lemmas, and to the next section. 9

10 Uniform asymptotics and the proofs of the lemmas In this section, we are going to prove Lemmas, and, using the method of uniform asymptotics ]. The method consists of two main steps. The first step is to transform the Lax pair equation. into a second-order Schrödinger equation, and to approximate the solutions of this equation with well known special functions. Indeed, denoting φ Φλ =, and defining Y λ, t = λ y φ, we have d Y dλ = y t + 4λ + λt yt 4y y t λ y + ] Y λ, t.. 4 λ y When t = 0, equation. is simplified to d Y dλ = 4λ + b 4a b λ a + ] Y λ.. 4 λ a One can regard. as either a scalar or a vector equation. We will see that in all cases I, II and III considered in this paper, the solutions of this Schrödinger equation can be approximated by certain special functions. In the second step, we can therefore use the known Stokes phenomena of these special functions. In each case, we shall calculate the Stokes multipliers of Y, and then calculate those of Φ. We carry out a case by case analysis to complete the steps.. Case I: fixed a and large positive negative b We may assume that b > 0. The argument for the case when b < 0 is the same, and hence omitted here. With the scaling λ = ξ η, b = ξ as ξ +, equation. is reduced to ] d Y dη = ξ 4η + η e πi πi 4a η e ξ ξ 6 ξ η a + ξ 6 Y 4ξ η a. := ξ F η, ξy, where F η, ξ = 4η + η e πi πi η e + gη, ξ,.4 ξη and gη, ξ = O ξ 6 as ξ +, uniformly for all η bounded away from η = 0. Obviously, there are three simple turning points, say η j, j = 0,,. Those are the zeros of F η, ξ near, e πi and e πi respectively. For convenience, we denote α = e πi and β = e πi. A straightforward calculation shows that η α and η 6ξ β as ξ +. 6ξ According to 7], the limiting state of the Stokes geometry of the quadratic form F η, ξdη as ξ + is described in Figure. Therefore, following the main ideas in ], we can 0 φ

11 γ γ γ η 0 η O l η γ 4 γ 0 Figure : Stokes geometry of F η, ξdη. approximate the solutions of. via the Airy functions. To do so, we define two conformal mappings ζη and ωη by ζ s ds = F s, ξ ds. 0 η and ω s ds = F s, ξ ds,.6 0 η respectively from neighborhoods of η = η and η = η to the origin. In the present paper, we take the principal branches for all the square roots. Then the conformality can be extended to the Stokes curves, and the following lemma is a consequence of, Theorem ]. Lemma 4. There are constants C, C and C, C, depending on ξ, such that ζ 4 Y = {C + o] Aiξ ζ + C + o] Biξ ζ, ξ +.7 F η, ξ uniformly for η on any two adjacent Stokes lines emanating from η ; and ω 4 { Y = C + o] Aiξ ω + C + o] Biξ ω, F η, ξ ξ +.8 uniformly for η on any two adjacent Stokes lines emanating from η. and Moreover, we get the asymptotic behavior of ζη and ωη as ξ, η +. Lemma. ζ 4 = η + E0 E ξ + o, arg η π ξ, π,.9 ω 4 = η + F0 F ξ + o, arg η π, π ξ as ξ, η + with η ξ, where E, F, E 0 and F 0 are constants stated in...0

12 The proof of Lemma is left to Appendix A. Now we turn to the proof of Lemma, assuming the validity of Lemma. Proof of Lemma. According to ], in order to calculate s 0, we should know the uniform asymptotic behaviors of Y on the two Stokes lines tending to infinity with arg η ± π. However, one may find that, in Figure, the two adjacent Stokes lines with arg η ± π, emanate from two different turning points. Hence we should build the relations between Aiξ ζ, Biξ ζ and Aiξ ω, Biξ ω. Actually, it can be done by matching them on the Stokes line l joining η and η. According to 8, 9.. and 9.7.], one has Aiz π z 4 e z, arg z π, π ; Aiz π z 4 e z + i π z 4 e z π, arg z, π ;. Aiz π z 4 e z i π z 4 e z π, arg z, π as z, and the uniform asymptotic behavior of Biz for arg z ± π by applying 8, 9..0], namely, can be evaluated Biz = e πi 6 Aize πi + e πi 6 Aize πi,. and the results are Biz z 4 e z + i π π z 4 e z, arg z π, π ; Biz z 4 e z i π π z 4 e z, arg z π, π ; Biz π z 4 e z + i π z 4 e z π, arg z, π. as z. Substituting. and.6 respectively into. and., we find that for η on the Stokes line l, ζ η 4 Aiξ ζ π ξ 6 exp { ξ F η, ξ ds ξ F η, ξ ds, η η ζ η 4 Biξ ζ π ξ 6 exp {ξ F η, ξ ds + ξ F η, ξ ds.4 η η i π ξ 6 exp { ξ F η, ξ ds ξ F η, ξ ds η η and ω η 4 Aiξ ω π ξ 6 exp { ξ F η, ξ ds, η ω η 4 Biξ ω π ξ 6 exp {ξ F η, ξ ds + i η π ξ 6 exp { ξ η F η, ξ ds.

13 as ξ +. Comparing.4 with., we have ζ 4 Aiξ ζ, Biξ ζ ω 4 Aiξ e Qξ i e Qξ + e Qξ] ω, Biξ ω 0 e Qξ.6 as ξ +, uniformly for η γ 0 γ l, where Qξ = ξ A straightforward calculation yields see Appendix B η F s, ξ ds..7 Qξ = iξq 0 Q + o as ξ +,.8 where πγ Q 0 = Γ = ie 0 F 0, Q = π = ie F..9 6 Now we begin to calculate s 0 via Φ λ = Φ 0 λs 0 and s via Φ λ = Φ λs. If η + with arg η π, then arg ζ π. Hence, substituting.9 into. and., and noting that λ = ξ η and the definition of F η, ξ in.4, we get ζ λ a F η, ξ ζ λ a F η, ξ as ξ, λ + with arg λ π, where 4 Aiξ ζ c λ 4 e 4 λ, 4 Biξ ζ ic λ 4 e 4 λ + c λ 4 e 4 λ.0 c = π ξ e ξe 0 +E +o and c = π ξ e ξe 0 E +o. as ξ +. In addition, a straightforward calculation from. leads to Φ k λ 4 e 4 λ and Φ k λ 4 e 4 λ, k Z, λ.. Comparing.0 with., and using the results of Lemma 4, we obtain Φ, Φ = ζ 4 λ a Aiξ ζ, Biξ F η, ξ ζ i c c c 0. as ξ +. Here, the c j s in. are not equal but asymptotically equal to the corresponding ones in. as ξ +. By abuse of notations, we use the same symbol for the c j s in these two formulas, since we only care about the asymptotic behavior of the Stokes multipliers.

14 If λ + with λ on the Stokes line γ 0, then arg ω π. Using a similar argument as in the derivation of., we get Φ 0, Φ 0 = ω 4 λ a Aiξ ω, Biξ F η, ξ ω i c c c 0,.4 where c = π ξ e ξf 0 +F +o and c = π ξ e ξf 0 F +o as ξ +. Combining.,.4 and.6, it is readily to get S 0 = = i c c c 0 e Qξ i e Qξ + e Qξ] i c c 0 e Qξ c 0. 0 i c c cos{ iqξ as ξ +. The last equality in. holds because Qξ = ξf 0 E 0 F E + o as ξ + ; cf..8 and.9. Finally, noting that E + F = 0 and.8, we get s 0 = ie ξe 0+F 0 +E +F +o cos { iqξ = ie ξe 0+F 0 +o cos {ξq 0 Q + o as ξ +. For the calculation of s, we need the asymptotic behaviors of the Airy functions as η, ξ + with arg η π and π, i.e. arg ζ π and π, which are given in. and.. Hence, following the way of deriving. or.4, we obtain Φ, Φ = ζ 4 i c λ a Aiξ ζ, Biξ ζ c..6 F η, ξ Combining. and.6, one can immediately get Φ, Φ = Φ, Φ Noting that the constants c and c are defined in., we then have s = i exp {ξe 0 E + o as ξ +. c i c i c c..7 0 Finally, other Stokes multipliers can be calculated via the constraint condition., and particularly, s = s = i + s 0 s = i exp {ξ E 0 F 0 E F + o.8 as ξ +. This completes the proof of Lemma. 4

15 . Case II: fixed b and large negative a In this case, it is appropriate to make the scaling a = ξ and λ = ξ η with ξ +, following which, equation. is reduced to ] d Y dη = ξ 4η + η e πi πi b b η e + + ξ 6 ξ 8 η + + Y 4ξ η +.9 := ξ F η, ξy, where F η, ξ = 4η + η e πi πi η e + gη, ξ.0 and gη, ξ = O ξ 6 as ξ +, uniformly for η bounded away from η =. Again there are three simple turning points, say η j, j = 0,, in the neighborhood of η =, η = α = e πi and η = β = e πi respectively. Moreover, we find that near the turning points η and η, equation.9 is similar to and even simpler than. in Case I, and the Stokes geometry of F η, ξdη is the same as the one shown in Figure. Hence, following the analysis in Subsection., we get { s0 = ie ξe 0+F 0 +o cos {irξ, s = ie ξe 0+o. as ξ +, where Rξ = ξ η η F s, ξ ds and E 0, F 0 are the same constants as the ones stated in.. Similar argument as the one of deriving.8 yields Rξ = iξq 0 + o. Therefore s 0 = ie a/ E 0 +F 0 +o cos { a / Q 0 + o as ξ +. Finally, other Stokes multipliers can be calculated via the constraint condition., and particularly, s = s = i + s 0 s = ie ξe 0 F 0 +o as ξ +.. This completes the proof of Lemma.. Case III: fixed b and large positive a With the scaling λ = ξ η and a = ξ, equation. is reduced to ] d Y dη = ξ 4η η e πi πi η e + b b + ξ 6 ξ 8 η + Y 4ξ η := ξ ˆF η, ξy,. where ˆF η, ξ = 4η η e πi πi η e + + ĝη, ξ,.4 4ξ η

16 and η ĝη, ξ = O ξ 6 as ξ +, uniformly for η bounded away from η =. Hence for large ξ there are η-zeros of ˆF η, ξ, known as turning points, near, ˆα = e πi and ˆβ = e πi, respectively. We note that the two turning points ˆηˆα and ˆη ˆβ near ˆα and ˆβ respectively are both simple and ˆηˆα ˆα = O ξ 6 and ˆη ˆβ ˆβ = O ξ 6.. Hence, in a neighborhood of each of these two turning points and on the stokes curves emanating form them, the Airy functions can also be used to uniformly approximate the solutions of.. Similar to. and.6, if we define χ then we have the following lemma which is similar to Lemma 4. 0 s ds = ˆF s, ξ ds,.6 ˆη ˆα Lemma 6. There are constants D and D, depending on ξ, such that Y = 4 χ {D + o] Aiξ χ + D + o] Biξ χ, ξ +,.7 ˆF η, ξ uniformly for η on two adjacent Stokes curves emanating from ˆηˆα. Moreover, the asymptotic behavior of χη as η, ξ + is stated as follows. Lemma 7. As ξ, η + with η ξ, where G 0 = B, 6 + i B, 6. χ 4 η + G0 + oξ, arg η π, π,.8 The proof of this lemma is similar to that of Lemma, and hence omitted here. Now we consider the uniform asymptotics of the solutions of. near η =. By a careful analysis, we find that there are three turning points ˆη j, j =,,, coalescing with the double pole η = of ˆF η, ξ as ξ +. Moreover, ˆη j e jπi 4 ξ as ξ +..9 According to the idea of 6], near η =, equation. can be approximated by the following differential equation d ϕ dδ = ξ δ + ] ϕ,.40 4ξ δ whose solutions can be represented by the modified Bessel functions as follows ϕ + δ = δ 4 I ξδ and ϕ δ = δ 4 K ξδ..4 6

17 Equation.40 also has three turning points, say δ j, j =,,, coalescing with the double pole δ = 0. More precisely, δ j e jπi 4 ξ for j =,, as ξ +. Hence, from.9, we see that δ j + ˆη j = oξ, j =,, as ξ +. Define δ δ s + ] η ds = ˆF s, ξ 4ξ s ds,.4 ˆη then δη is a conformal mapping in the neighborhood of η =. following lemma. Moreover, we have the Lemma 8. There are two constants A and A, depending on ξ, such that Y = ] δ + 4 4ξ δ {A + o] ϕ + δ + A + o] ϕ δ.4 ˆF η, ξ uniformly for η on any two adjacent Stokes lines emanating from ˆη j, j =,,. The proof of this lemma is similar to those of, Theorems and ]. By evaluating the integrals in.4 see Appendix C, we get the asymptotic behavior of δη as follows. Lemma 9. As ξ, η + with η ξ, the asymptotic behavior of δη is given by where H 0 = B 6,. 4 δ 4 = η H0 + oξ, arg η π, π,.44 Now we turn to the proof of Lemma, i.e., to derive the asymptotic behavior of the Stokes multipliers s k as ξ +, k = 0,,,, 4. Proof of Lemma. To calculate s 0, we use the uniform asymptotics of the solutions of. near η = ; see Lemmas 8 and 9. If arg η π as η, then arg δ π. Recall the uniform asymptotic behaviors of the modified Bessel functions stated in 8, 0.4., 0.40.] and λ = ξ η. Then.4 and.44 lead to ] δ + 4 4ξ λ a δ ϕ + δ f ˆF η, ξ λ 4 ] δ + 4 4ξ λ a δ ˆF η, ξ as η, ξ + with arg η π, where ϕ δ f λ 4 e 4 λ + f e 4 λ, λ 4 e 4 λ.4 f = /4 π ξ e ξh 0 +o and f = i/4 e πi π ξ e ξh 0 +o = e πi π f,.46 7

18 as ξ +. Comparing.4 with., we get Φ, Φ = λ a ] 4δ + 4 4ξ δ ϕ + δ, ϕ δ ˆF η, ξ f f f f f If arg η π as η +, then arg δ π. Using a similar argument for deriving.47, one has Φ 0, Φ 0 = ] 4δ ξ λ a δ ˆf ϕ + δ, ϕ δ,.48 ˆF η, ξ ˆf ˆf ˆf ˆf where ˆf = f, ˆf = e 4πi f and ˆf = f. Combining.47 and.48, we have 0 S 0 = ˆf f f..49 Therefore s 0 = ˆf f f = ie ξh0+o as ξ +. Next, to derive s, we need the uniform asymptotics of the solutions of. near η = ˆα see Lemmas 6 and 7. Following the analysis of deriving s in case I or II, we obtain s = i exp {ξg 0 + o, as ξ +..0 Other Stokes multipliers can be derived by the constraint condition. and.6, and particularly, s = s = i + s 0 s = i exp {ξ G 0 + H 0 + o. and as ξ +. This completes the proof of Lemma. 4 Discussion s = s = i exp { ξg 0 + o. We have considered the initial value problem of. with initial data.4 in three special cases. In fact we have given a rigorous proof of the conclusions in ] for PI, built more precise relations between the initial value of PI solutions and their large negative t asymptotic behavior, and obtained the limiting form connection formulas..8. There are still several issues to be further investigated. First, we have only considered three special cases of the initial value problem of PI. An equally natural question is what would happen when both a and b are large. More attention should be paid to 9, Fig. 4.] by Fornberg and Weideman. According to this figure, we find that it may be divided into two cases: large b and large negative a, large b and large positive a. 4. 8

19 Moreover, we see that the analysis of the first case in 4. is similar to one of case I or case II in the present paper, while for the second case in 4., more careful analysis is needed. For example, a description of the Stokes geometry and the Stokes curves at the finite plane seems to be vital; cf. ]. Of course, one may consider other special cases of Clarkson s open problem. For example y0 = y 0 = 0. According to ], see also in 9], the PI solution with y0 = y 0 = 0 oscillates about y = t 6 when t < 0, and it satisfies. and.. Hence, a question is how to determine the exact values of the parameters d and θ in. and.. In fact, the result is already known, and is given by Kitaev 6] via the WKB method. To the best of our knowledge, it may be the only special case of the Clarkson s open problem which has been fully solved for PI equation. However, a more important and challenging problem is the general case of Clarkson s open problem. Assuming that a and b are fixed parameters, then equation. has a regular singularity at λ = a and an irregular singularity at λ = of rank. As far as we know, there is no such known special function that can be used to approximate the solutions of., uniformly for λ near both a and. This is the main difficulty of the general case of Clarkson s open problem. Finally, it seems quite promising that the method of uniform asymptotics can also be applied to the initial value problems of other Painlevé equations. In fact, similar results for the properties of the PII solutions have also been stated in ]. Acknowledgements The work of Wen-Gao Long was supported in part by the National Natural Science Foundation of China under Grant Number 776 and GuangDong Natural Science Foundation under Grant Number 04A0076. The work of Yu-Tian Li was supported in part by the Hong Kong Research Grants Council grant numbers 0 and 0] and the HKBU Strategic Development Fund. The work of Sai-Yu Liu was supported in part by the National Natural Science Foundation of China under grant numbers 608 and The work of Yu- Qiu Zhao was supported in part by the National Natural Science Foundation of China under grant numbers 087 and 77. Appendix A: Proof of Lemma The idea to prove Lemma is to derive the asymptotic behavior of the integral on the right hand side of. as ξ, η +. Here we only show the validity of.9. The proof of.0 is similar and hence omitted here. Choose T to be a point such that T η ξ 4, then T F s, ξ ds = F s, ξ ds + F s, ξ ds := I + I. A. η η T 9

20 Noting that F η, ξ = Oξ 4 as ξ + uniformly for η on the integration contour from η to T, we immediately conclude that I = oξ. Since T η ξ 4 and η α 6ξ as ξ + see the paragraph under.4, then T α > c ξ 4 for some constant c > 0. By the Taylor expansion of F s, ξ with respect to, we have ξ F s, ξ =s + s α s β ξss + s α s β + O s + s αs β ξ 6 A. as ξ +, uniformly for s on the integration contour from T to η, where α = β = e πi. Therefore, I = = T s + s α s β ds T s + s α s β ds α T ξss + s α s β T ξss + s α s β ds + o ξ ds + o ξ provided that η as ξ +. Recalling the facts that T η ξ 4 and η α, 6ξ we further obtain I = s + E s α s β ds ξ + o, A. ξ where E = α α ss + s α s β Next, using integration by parts, we have s + s α s β ds = E0 + 4 η + o ξ as ξ, η + provided that η ξ and arg η π, π, where E 0 = 6 s + ds = B, i B α ds = πi 6. A.4, Finally, combining A., A. and A., we obtain the desired formula.9. Appendix B: Proof of.8 and.9 A.. A.6 First, similar to the proof of Lemma, we choose T and T such that Tj η j ξ 4 for j =, as ξ +. Then we split the integral in.7 to give Qξ = ξ T F s, ξds + ξ T F s, ξds + ξ F s, ξds := K + K + K. B. η T T 0

21 By the same argument of estimating I in Appendix A, we have K, K = o as ξ +. Next, we estimate K. Making use of the facts η α, η 6ξ β see the paragraph 6ξ under.4 and Tj η j ξ 4 as ξ +, we conclude that T α ξ 4 and T β ξ 4. B. Then, by the Taylor expansion of F s, ξ with respect to, we get ξ F s, ξ =s + s α s β + ξss + s α s β + O s + s αs β ξ 6 B. as ξ + uniformly for s on the integration contour from T to T. Hence, K = ξ T T F s, ξ ds =ξ T T T T s + s α s β ds ds + o. s + s α s β s Using B. again, we can further obtain K =ξ α β α s + s α s β ds β ds + o s + s α s β s B.4 as ξ +. Therefore, combining B. and B.4, we get Qξ = iξq 0 Q + o as ξ +, B. where Q 0 = i Q = i α β α β s + s αs βds = πγ Γ ds s + s βs αs = π. = B 6,, B.6 Finally, in view of the explicit representations. of E 0, F 0, E and F, we see that F E = iq and F 0 E 0 = iq 0. Appendix C: Proof of Lemma 9 The idea to prove Lemma 9 is to derive an asymptotic approximation for each integral in.4.

22 Choose δ to be a point with δ δ ξ as ξ +, then δ δ s + ] δ ds = s + ] δ ds + s + ] ds := J + J 4ξ s δ 4ξ s δ 4ξ s. C. Similarly, we split the integral on the right-hand side of.4 as follows +δ ˆF s, ξ ds = ˆF s, ξ ds + ˆF s, ξ ds := Ĵ + Ĵ. C. ˆη ˆη +δ Since when s is on the integration contour from η to + δ, as ξ +, then ˆF s, ξ ] Ĵ = δ ˆη = = s + s + 4ξ s + Oξ 4ξ s ] ] 4 + Oξ C. s + ] ds + Oξ 4ξ s 0 = J + oξ C.4 as ξ +. Here, we have made use of the fact that ˆη δ = oξ, mentioned in the paragraph above.4. Combining C., C. and C.4, we find that.4 is reduced to J = Ĵ + oξ as ξ +. C. Now we turn to the estimation of J and Ĵ. Since ξ s = o as ξ +, uniformly for s on the integration contour from δ to δ, then s + ] = s 4ξ s + O ξ s as ξ +. C.6 Hence, we have J = 4 δ 4 δ + oξ as ξ +. C.7 Similarly, when s is on the integration contour from δ + to η, ˆF s, ξ = s s α / s β + O as ξ +, C.8 ξ s since δ ξ, as can be derived from the facts δ e jπi 4 ξ see the paragraph under.4 and δ δ ξ. Then, we have Ĵ = = δ + s s α s β ds + oξ δ + s ds s C.9 ds + o ξ

23 as ξ +. Using the fact δ ξ again, a simple calculation yields Therefore, δ + s ds = 4 δ + o ξ, as ξ. Ĵ = s ds 4 δ + o ξ = 4 η H0 4 δ + o ξ, where C.0 as ξ, η +, provided that η ξ and arg η π, π H 0 = 6 s ds = B 6,. C. A combination of C., C.7 and C.0 leads to.44. This completes the proof of Lemma 9. Appendix D: Calculation of integrals in A.6 and B.6 To verify the first formula in B.6, it suffices to show that α i I = s + s αs βds = B,, D. β where α = e πi/, β = e πi/, the principal branches in the integrand are chosen such that args +, args α, args β π, π, and the integration path is the line segment connecting β and α, subject to deformation. It is readily seen that I = α s + ds. Integrating by parts once, we have β β I = I + α Applying Cauchy s integral theorem, we further have I = α ds s + = e πi/ β β ds s +. ds e πi/ s + α ] ds, s + where, in the last integrals, we may take the integration paths to be portions of the rays emanating from the origin. Now a change of variable t = se ±πi/ gives I = i + dt t = i B, = i 6 B,. Here, in the last equality, use has been made of the reflection formula of the gamma function. Thus we have D., and the formula in B.6 for Q 0 follows accordingly.

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